How can I move water from down to up through a motor?

You can install a submersible pump inside the tank and run the water line out near the top of the tank adjacent to the overflow. This will require electrical work. Or if you have an external pump already, you could simply run the intake line into the barrel and let the pump suck it it out. Consider installing a filter in-line before the pump on any application as well as a screen over the rainwater collection hole. Also check to see what size pump motor is needed to push your water where it needs go at the rate you wish it to flow and the pump electrical requirements.